The object in this case is a direct object. Ask yourself, "What/Whom do you believe?" If you can answer "him, her or you" (or their plurals), you are dealing with a direct object.
The direct object pronoun for 'him' is 'lo'. For 'her', it is 'la'. For 'you (formal-Usted)', first determine the gender and then use 'lo' or 'la', accordingly.

I will have to disagree about "leísmo" and "loísmo" here: When you say "don't believe her/him", we say "no le creas" (Indirect Object), because in Spanish we believe things, not people themselves and we are rather talking about the thing they say, which we assume is false.
But we say "no lo/la creas" (Direct Object) when it's the thing itself that mustn't be believed.
So when talking about a person, it's "loísmo", rather than "leísmo" what sounds strange.

- Los niños dicen que tú les diste dulces; no les creo.
- The children say you gave them some candy; I don't believe (it-mandatory English direct object pronoun) (of) them.
-or- I don't believe (them when they say) (it-mandatory English direct object: (that) you gave them some candy).